My iMac refuses to start after a power cut

It would help if you provide more info about your iMac. They have been in continuous production since 1998 in four broad families with different troubleshooting procedures. Also there are literally hundreds of iMac variants. This forum covers iMacs in 1998-2005. If the computer is not running and you don't remember the sub-model, you can find it by serial number from this Apple page:

https://checkcoverage.apple.com

That link ofter may fail to work witb older Macs. In that case, you can find the same model reporting function here:

http://www.everymac.com/ultimate-mac-lookup/

If you are not sure where to find the serial number, please see this Apple article:

How to find the serial number on your iMac - Apple Support

My suspicion would be that it was damaged due to a power spike when the power was restored. You may want to have it examined at an Apple Store or Apple Authorized Service provider. Also note that the damage may be covered by your homeowners insurance.

As I noted, it is only my suspicion. Major power spikes can damage electronic equipment even when off. My brother had major damage to appliances and computers all of which were off. In his case his local power utility compensated him as it was a problem that they caused.

Have you done the simple things?

Verify the socket/ wall outlet is still working with other devices?

Check the fuse in the mains cable to the iMac (obviously this varies by country)?

Unplug the iMac for a few hours to allow it to fully discharge & see if it starts up when reconnecting to the outlet.

Disconnect all peripherals too except the keyboard and mouse.

Hello Alan Mather,

In my opinion FoxFifth is right-- the power outage damaged your computer. I have a 2016 iMac, and the heavy snow on the power lines in December gave us split-second power outages and damaged my hard drive. Fortunately, it's still under warranty, so Apple did not charge me for the replacement. We immediately purchased a UPS (uninterruptible power supply) to protect our computers at home. Thanks FoxFifth, I'm planning to write our local power utility, because my husband had to purchase a new hard drive for his PC as well. Also, the blower fan of our fireplace insert died last week, and I believe it was the intermittent power outages in our area that killed it.